A fascinating approach to the Fresnel integrals

Back with the Fresnel integrals but this time we're evaluating them using an approach you've probably never seen before: using the wonderful gamma function.

[illumexhisoka]

At last
A reasonable solution that I can understand
I have been trying to know a know how to do these for months but since I am a high school student most solutions were either or too hard for me(since when I learn something I want to know everything from the scratch)
I understood everything except your solution of the √tan(x) integral
Luckily you posted a video 2 months ago with a simpler solution
Thank you for the video

[rajendramisir]

Good use of the Beta and Gamma integrals. Impressive choice of substitutions and understanding of the relationships between these tools.

[tsa_gamer]

Does anyone remember the Gaussian integral?
0 to infinity integration of e ^(-ax^2)
so -a=i
we know that this integral is equal to 1/2 x sqrt(pi/a)
so put a=-i
we get our result to be 1/2 x sqrt (pi/(-i))= 1/2 x sqrt(ipi) = [sqrt (pi)/2 ] x sqrt(i) and we all know what is square root of i

[rickster]

Hi sir, great video as always. However, I am a bit confused at 4:23. I understand how you got there, but isn't (1/ √x ) for the parameter value not the integration variable, right? Can you interchange them like that?

[LeaoDN]

Thanks for the hard work. BTW, I don't understand the xt substitution. It looks like the integral expression on 3:33 under variable u is the same as the expression under t on 2:58. That would imply that 1/x^1/2 had to be equal to one which it is obviously not the case. Am I missing something? Again, thanks.

[jaafars.mahdawi]

Great indirect method to solve these innocent-looking integrals. Just one question, when cot was turned to tan, isn't there a missing minus sign due to the reversion of the boundaries? I must be mistaken as the result seems correct, but can't see how. Thank you.

[manstuckinabox]

I heard it slightly involved complex numbers, ran as fast as I can.
11:18 well you know what they say, "small mistakes makes rockets crash to planets, or a heap of math nerds really angry."

[illumexhisoka]

I tried to come up with a general formula for the power is n(any positive integer) instead of 2 using your method
But I stopped at the last step
Integrate tan^a(x) from 0 to π÷2
Is there an easy way to do this?

[cameronspalding]

I would approach this buy integrating exp(i x^2) over a limiting contour which in this case would be a sector of radius R>0 and angle pi/4, let R tend to infinity so that the contribution from the circular arc goes to zero

[samueljele]

Question: when you evaluate the limits, you say that when x->infinity, exp(-x*(t-i)) goes to zero. The outer integral w.r.t. t goes from 0 to infinity. The limit above does not converge when t=0. Is it ok to say the case where t=0 is excluded in the outer integral because you can always remove null-sets from integration domains and still get the same result, i.e. the integral over t>=0 and t>0 are the same?

[giacomocervelli]

Another way to do it, its by studying
I(t)=(integral from zero to infinity)exp(-tx^2)
the feynmann way. Then you substitute t=i and you evaluate the real and imaginary parts

[bobtannous]

If you take a change of variable the equal the equal x √ small I

[memeboy]

You didn’t have to do all of that. Suppose that the parametrised gamma function integral is denoted as J. Which you call I_1 (i will call it J). The value of J(x) = sqrt(π) / sqrt(x)

Our original intgral I is just 1/2 ( J(i) )
We can just substitute x = i in the above formula and get the answer! There is no need to do a double integral other than to flaunt you integration skills of course!

[Aditya_]

11:19 💀 i thought bro messed up something too important huh it was just a constant but really I was scared

[Count.Saruman]

Oh, I've got fond memories of this one. Back in high school I tried to do it without complex analysis for some reason (probably a stupid bet over a drink) and ended up spending a week on it. In the end the solution ended up with another well-known integral of dx/(x⁴ + 1), which incidentally had been taught in that very same course...

[illumexhisoka]

it took me a while to find a simple video that proof the relationship between the gamma and the Betta functions but I fainaly found it

The integral from 0 to ∞ of
e^(i(x^a))(when a is any positive number greater than 1) is equal to



Not the simplest way to write it but this is the best I could

I wrote it in terms of the factorial function instead of the gamma function because it Confuses me sometimes

It surprised me that if the power changed the values of the two integrals won't be the same any more

I am not sure if the formula work with complex numbers
When I substituted with I in the formula it gave me a finite value but when I computed the integral in Wolfram alpha to compare the answers it didn't give me anything

Again thank you for the video